Expansion of the correlation functions of the grand canonical ensemble in powers of the activity

A study is made of the grand canonical ensemble of single-component systems of particles in a region $\Lambda$. A new representation of the Ursell functions is given. In it an Ursell function is represented as a sum of products of Mayer and Boltzmann functions over the subset of connected graphs labeled by trees. Such a representation greatly reduces the complexity of the structure of these functions. A new definition of all-round tending of the region $\Lambda$ to infinity is given. The relationship between this definition and the well-known definition of tending of the set $\Lambda$ to infinity in the sense of Fisher is demonstrated in examples. It is shown that in the case of all-round tending of the set $\Lambda$ to infinity a term-by-term passage to the limit can be made in the series in Ruelle's representation of the correlation functions as a finite sum of finite products of convergent series. The domain of convergence of the obtained expansions is discussed. As examples, the expansions of the single-particle and binary correlation functions are obtained.